PHYSICAL REVIEW E, VOLUME 63, 056606
Shear representations of beam transfer matrices
S. Bas¸kal* Department of Physics, Middle East Technical University, 06531 Ankara, Turkey
Y. S. Kim† Department of Physics, University of Maryland, College Park, Maryland 20742 ͑Received 25 August 2000; published 13 April 2001͒ The beam transfer matrix, often called the ABCD matrix, is one of the essential mathematical instruments in optics. It is a unimodular matrix whose determinant is 1. If all the elements are real with three independent parameters, this matrix is a 2ϫ2 representation of the group Sp͑2͒. It is shown that a real ABCD matrix can be generated by two shear transformations. It is then noted that, in para-axial lens optics, the lens and translation matrices constitute two shear transformations. It is shown that a system with an arbitrary number of lenses can be reduced to a system consisting of three lenses.
DOI: 10.1103/PhysRevE.63.056606 PACS number͑s͒: 42.79.Ci, 11.30.Cp, 02.20.Qs
I. INTRODUCTION light ͓4͔. This group is also applicable to other branches of optics, including polarization optics, interferometers, layer In a recent series of papers ͓1,2͔, Han et al. studied pos- optics ͓5͔, and para-axial optics ͓6,7͔. The Sp͑2͒ symmetry sible optical devices capable of performing matrix operations can be found in many other branches of physics, including of the following types: canonical transformations ͓3͔, special relativity ͓4͔, Wigner functions ͓4͔, and coupled harmonic oscillators ͓8͔. 1 a 10 This group, consisting of 2ϫ2 real matrices, also has a Tϭͩ ͪ , Lϭͩ ͪ . ͑1.1͒ 01 b 1 very rich mathematical contents. There are still hidden math- ematical theorems which can be useful in managing our cal- Since these matrices perform shear transformations in a two- culations in physics. Specifically, we use group theory to dimensional space ͓3͔, we shall call them ‘‘shear’’ matrices. represent the most general form of the ABCD matrix in However, Han et al. were only interested in the ‘‘slide-rule terms of the shear matrices given in Eq. ͑1.1͒. With this point property’’ of the shear matrices which convert multiplica- in mind, we propose to write the 2ϫ2 ABCD matrices in tions into additions. The T matrix has the property the form ϩ 1 a1 1 a2 1 a1 a2 TLTLT ... . ͑1.4͒ T T ϭͩ ͪͩ ͪ ϭͩ ͪ ͑1.2͒ 1 2 01 01 01 Since each matrix in this chain contains one parameter, there and the L matrix has a similar ‘‘slide-rule’’ property. This are N parameters for N matrices in the chain. On the other property is valid only if we restrict computations to T-type hand, since both T and L are real unimodular matrices, the matrices or L-type matrices. final expression is also real and unimodular. This means that In the present paper, we study what happens to the ABCD the expression contains only three independent parameters. matrix if we use both L- and T-type shear matrices, which Then we are led to the question of whether there is a shortest takes the form chain which can accommodate the most general form of the 2ϫ2 matrices. We shall conclude in this paper that six ma- AB trices are needed for the most general form, with three inde- Gϭͩ ͪ , ͑1.3͒ CD pendent parameters. We are not the first ones to study this problem. In 1985, where the elements A, B, C, and D are real numbers satis- Sudarshan et al. raised this question in connection with para- fying ADϪBCϭ1. Because of this condition, there are three axial lens optics ͓7͔. They observed that the lens and trans- independent parameters. lation matrices are in the form of matrices given in Eq. ͑1.1͒. We are interested in constructing the most general form of In fact, the notations L and T for the shear matrices of Eq. the ABCD matrix in terms of the two shear matrices given in ͑1.1͒ are derived from the words ‘‘lens’’ and ‘‘translation,’’ Eq. ͑1.1͒.2ϫ2 matrices with the above property form the respectively, in para-axial lens optics. Sudarshan et al. con- symplectic group Sp͑2͒. We are quite familiar with the con- cluded that three lenses are needed for the most general form ventional representation of the 2ϫ2 representation of the for the 2ϫ2 matrices for the symplectic group. Of course Sp͑2͒ group. This group is like ͑isomorphic to͒ SU͑1,1͒ their lens matrices are appropriately separated by translation which is the basic scientific language for squeezed states of matrices. This will make the total number of matrices six. However, in their paper Sudarshan et al. stated that the cal- culation of each lens or translation parameter is ‘‘tedious.’’ *Electronic address: [email protected] In the present paper, we make this calculation less tedious †Electronic address: [email protected] by using a decomposition of the ABCD matrix derivable
1063-651X/2001/63͑5͒/056606͑6͒/$20.0063 056606-1 ©2001 The American Physical Society S. BAS¸KAL AND Y. S. KIM PHYSICAL REVIEW E 63 056606 from Bargmann’s paper ͓9͔. As far as the number of lenses is ͓7͔ studied this problem in connection with para-axial lens concerned, we reach the same conclusion as that of Sudar- optics. Their approach was of course correct, however they shan et al. However, we complete the calculation of lens concluded that the complete calculation is ‘‘tedious.’’ parameter for each lens and the translation parameter for We propose to complete this well-defined calculation by each translation matrix, in terms of the three independent decomposing the matrix G into one symmetric matrix and parameters of the ABCD matrix. In other words, we com- one orthogonal matrix. For this purpose, let us consider the plete the calculation which Sudarshan et al. started in 1985. most general form of an Sp͑2͒ matrix by referring to Appen- In Sec. II, it is shown that the most general form of the dix B. It is shown that the matrix G can be written as Sp͑2͒ matrices or ABCD matrices can be decomposed into cos Ϫsin e 0 cos Ϫsin one symmetric matrix and one orthogonal matrix. It is shown ϭͩ ͪͩ ͪͩ ͪ G Ϫ , that the symmetric matrix can be decomposed into four shear sin cos 0 e sin cos matrices and the orthogonal matrix into three. In Sec. III, it is ͑2.1͒ noted that the mathematical device developed in Sec. II is directly applicable to para-axial lens optics. It is shown that where the three free parameters are , , and . These ma- the most general lens system can be reduced to six shearlike trices are generated from the squeeze representation of Sp͑2͒ matrices with three lens matrix. In Sec. IV, we discuss other given in Eq. ͑A6͒. The real numbers A, B, C, and D in Eq. areas of optical sciences where the shear representation of ͑1.3͒ can be written in terms of these three parameters. Con- the group Sp͑2͒ may serve useful purposes. We also discuss versely, the parameters , , and can be written in terms of a possible extension of the ABCD matrix to a complex rep- A, B, C, and D, with the condition that ADϪBCϭ1. This resentation, which will enlarge the group Sp͑2͒ to a larger matrix is written in terms of squeeze and rotation matrices. group. We write the last matrix of Eq. ͑2.1͒ as Even though we are dealing with real 2ϫ2 matrices, we Ϫ are using mathematical operations not too familiar in optics. cos sin cos sin ͩ ͪͩ ͪ , ͑2.2͒ In Appendix A, we show that there are two different ways of Ϫsin cos sin cos representing the Sp͑2͒ group. One is a conventional repre- sentation combining squeeze and rotations. The other is to with ϭϪ. Instead of , becomes an independent pa- use two shears and one squeeze. It is shown that these are rameter. equivalent. In Appendix B, we use the Bargmann decompo- The matrix G can now be written as two matrices, one sition ͓9͔ to prove that the most general form of the ABCD symmetric and the other orthogonal, matrix can be written as a product of one symmetric matrix ϭ ͑ ͒ and one antisymmetric matrix. G SR, 2.3 with II. DECOMPOSITIONS AND RECOMPOSITIONS Ϫ In this paper we are interested in writing the most general cos sin Rϭͩ ͪ . ͑2.4͒ form of the matrix G of Eq. ͑1.3͒ as a chain of the shear sin cos matrices. It is shown in Appendix A that this is possible in terms of the generators of the Sp͑2͒ group. Sudarshan et al. The symmetric matrix S takes the form ͓2͔
cosh ϩ͑sinh ͒cos͑2͒ ͑sinh ͒sin͑2͒ Sϭͩ ͪ . ͑2.5͒ ͑sinh ͒sin͑2͒ cosh Ϫ͑sinh ͒cos͑2͒
Our procedure is to write S and R separately as shear chains. 101 Ϫsin 10 RЈϭͩ ͪͩ ͪͩ ͪ . Let us first consider the rotation matrix. tan͑/2͒ 1 01tan͑/2͒ 1 In terms of the shears, the rotation matrix R can be written ͑2.7͒ as ͓10͔. Both R and RЈ are the same matrix, but are decomposed in 1 Ϫtan͑/2͒ 101 Ϫtan͑/2͒ different ways. Rϭͩ ͪͩ ͪͩ ͪ . As for the two-parameter symmetric matrix of Eq. ͑2.5͒, 01sin 1 01 we start with a symmetric LTLT form ͑2.6͒ 10 1 a 10 1 b Sϭͩ ͪͩ ͪͩ ͪͩ ͪ , ͑2.8͒ This expression is in the form of TLT, but it can also be b 1 01 a 1 01 written in the form of LTL. If we take the transpose and change the sign of , R becomes which can be combined into one symmetric matrix:
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1ϩa2 b͑1ϩa2͒ϩa III. PARA-AXIAL LENS OPTICS Sϭͩ ͪ . ͑2.9͒ b͑1ϩa2͒ϩa 1ϩ2abϩb2͑1ϩa2͒ So far, we have been investigating the possibilities of rep- resenting the ABCD matrices in terms of the two shear ma- By comparing Eqs. ͑2.5͒ and ͑2.9͒, we can compute the pa- trices. Indeed, this ABCD matrix has a deep root in ray rameters a and b in terms of and . The result is optics ͓6͔. In para-axial lens optics, the lens and translation matrices ϭϮͱ͑ Ϫ ͒ϩ͑ ͒ ͑ ͒ a cosh 1 sinh cos 2 , take the forms ͑sinh ͒sin͑2͒ϯͱ͑cosh Ϫ1͒ϩ͑sinh ͒cos͑2͒ bϭ . 10 1 s cosh ϩ͑sinh ͒cos͑2͒ Lϭͩ ͪ , Tϭͩ ͪ , ͑3.1͒ Ϫ1/f 1 01 ͑2.10͒ This matrix can also be written in a TLTL form: respectively. In Sec. I, this was what we had in mind when we defined the shear matrices of L and T types. These ma- 1 bЈ 101 aЈ 10 trices are applicable to the two-dimensional space of SЈϭͩ ͪͩ ͪͩ ͪͩ ͪ . ͑2.11͒ 01 aЈ 1 01 bЈ 1 y ͩ ͪ , ͑3.2͒ Then the parameters aЈ and bЈ are m ЈϭϮͱ͑ Ϫ ͒Ϫ͑ ͒ ͑ ͒ a cosh 1 sinh cos 2 , where y measures the height of the ray, while m is the slope of the ray. ͑sinh ͒sin͑2͒ϯͱ͑cosh Ϫ1͒Ϫ͑sinh ͒cos͑2͒ bЈϭ . The one-lens system consists of a TLT chain. The two- cosh Ϫ͑sinh ͒cos͑2͒ lens system can be written as TLTLT. If we add more lenses, ͑2.12͒ the chain becomes longer. However, the net result is one ABCD matrix with three independent parameters. In Sec. II, The difference between the two sets of parameters ab and we asked the question of how many L and T matrices are aЈbЈ is the sign of the parameter . This sign change means needed to represent the most general form of the ABCD that the squeeze operation is in a direction perpendicular to matrix. Our conclusion was that six matrices, with three lens the original direction. In choosing ab or aЈbЈ, we will also matrices, are needed. The chain can be either LTLTLT or have to take care that the sign of the quantity inside the TLTLTL. In either case, three lenses are required. This con- square root is positive. If cos(2) is sufficiently small, both clusion was obtained earlier by Sudarshan et al. in 1985 ͓7͔. sets are acceptable. On the other hand, if the absolute value In this paper, using the decomposition technique derived of (sinh )cos(2) is greater than (cosh Ϫ1), only one of from the Bargmann decomposition, we were able to compute the sets, ab or aЈbЈ, is valid. the parameter of each shear matrix in terms of the three We can now combine the S and R matrices in order to parameters of the ABCD matrix. construct the ABCD matrix. In so doing, we can reduce the In para-axial optics, we often encounter special forms of number of matrices by one: the ABCD matrix. For instance, the matrix of the form of ͑ ͒ ͓ ͔ 10 1 a 10 1 bϪtan͑/2͒ Eq. A4 is for pure magnification 11 . This is a special case SRϭͩ ͪͩ ͪͩ ͪͩ ͪ of the decomposition given for S and SЈ in Eqs. ͑2.9͒ and b 1 01 a 1 01 ͑2.11͒ respectively, with ϭ0. However, if is positive, the 101 Ϫtan͑/2͒ set aЈbЈ is not acceptable because the quantity in the square ϫͩ ͪͩ ͪ . ͑2.13͒ ͑ ͒ sin 1 01 root in Eq. 2.12 becomes negative. For the ab set, the parameters are related by We can also combine making the product SЈRЈ. The result is aϭϮ͑eϪ1͒1/2, bϭϯeϪ͑eϪ1͒1/2. ͑3.3͒ 1 bЈ 101 aЈ 10 ͩ ͪͩ ͪͩ ͪͩ ͪ ͑ ͒ 01 aЈ 1 01 bЈϩtan͑/2͒ 1 The decomposition of the LTLT type is given in Eq. 2.8 . We often encounter the triangular matrices of the form 1 Ϫsin 10 ͓ ͔ ϫͩ ͪͩ ͪ . ͑2.14͒ 12 01 tan͑/2͒ 1 AB A 0 For the combination SR of Eq. ͑2.13͒, two adjoining T ma- ͩ ͪ or ͩ ͪ . ͑3.4͒ 0 D CD trices were combined into one T matrix. Similarly, two L matrices were combined into one for the SЈRЈ combination of Eq. ͑2.14͒. However, from the condition that their determinant be 1, In both cases, there are six matrices, consisting of three T these matrices take the form matrices and three L matrices. This is the minimum number e B e 0 of shear matrices needed for the most general form for the ͩ ͪ or ͩ ͪ . ͑3.5͒ ABCD matrix with three independent parameters. 0 eϪ CeϪ
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The first and second matrices are used for focal and telescope Indeed, this six-parameter group can accommodate a wide conditions, respectively. We call them the matrices of B and spectrum of optics and other sciences. Recently, the 2ϫ2 C types, respectively. The question then is how many shear Jones matrix and 4ϫ4 Mueller matrix were shown to be 2 matrices are needed to represent the most general form of ϫ2 and 4ϫ4 representations of the Lorentz group ͓1͔. Also these matrices. The triangular matrix of Eq. ͑3.4͒ is dis- recently, Monzo´n and Sa´nchez showed that multilayer optics cussed frequently in the literature ͓11,12͔. In the present pa- could serve as an analog computer for special relativity ͓5͔. per, we are interested in using only shear matrices as ele- More recently, it was noted that two-beam interferometers ments of decomposition. can also be formulated in terms of the Lorentz group ͓17͔. Let us consider the B type. It can be constructed either in the form V. CONCLUDING REMARKS Ϫ e 0 1 e B The Lorentz group was introduced to physics as a math- ͩ ͪͩ ͪ ͑3.6͒ 0 eϪ 01 ematical device to deal with Lorentz transformations in spe- cial relativity. However, this group is becoming the major or language in optical sciences. With the appearance of squeezed states as two-photon coherent states ͓18͔, the Lor- 1 eB e 0 ͩ ͪͩ ͪ . ͑3.7͒ entz group was recognized as the theoretical backbone of 01 0 eϪ coherent states as well as generalized coherent states ͓4͔. In their recent paper ͓2͔, Han et al. studied in detail pos- The number of matrices in the chain can be either four or sible optical devices which produce the shear matrices of Eq. five. We can reach a similar conclusion for the matrix of the ͑1.1͒. This effect is due to the mathematical identity called C type. ‘‘Iwasawa decomposition’’ ͓19,20͔. The shear matrices of Eq. ͑1.1͒ are products of Iwasawa decompositions. Since we IV. OTHER AREAS OF OPTICAL SCIENCE used those shear matrices to produce the most general form of 2ϫ2 unimodular matrices with three real parameters, in We write the ABCD matrix for the ray transfer matrix this paper we are performing an inverse process of the ͓11͔. There are many ray transfers in optics other than para- Iwasawa decomposition. axial lens optics. For instance, a laser resonator with spheri- It should be noted that the decomposition we used in this cal mirrors is exactly like para-axial lens optics if the radius paper has a specific purpose. If purposes are different, dif- of the mirror is sufficiently large ͓13͔. ferent forms of decomposition may be employed. For in- If wave fronts with phase are taken into account, or for stance, decomposition of the ABCD matrix into shear, Gaussian beams, the elements of the ABCD matrix become squeeze, and rotation matrices could serve useful purposes complex ͓14,15͔. In this case, the matrix operation can some- for canonical operator representations ͓12,21͔. The amount times be written as of calculation seems to depend on the choice of decomposi- AwϩB tion. wЈϭ , ͑4.1͒ Group theory in the past was understood as an abstract CwϩD mathematics. In this paper, we have seen that it can be used where w is a complex number with two real parameters. This as a calculational tool. is precisely the bilinear representation of the six-parameter Lorentz group ͓9͔. This bilinear representation was discussed APPENDIX A: SQUEEZE AND SHEAR in detail for polarization optics by Han et al. ͓16͔. This form REPRESENTATIONS OF THE Sp„2… GROUP of representation is also useful in laser mode-locking and optical pulse transmission ͓15͔. The ABCD matrix is a shear representation of the group The bilinear form of Eq. ͑4.1͒ is equivalent to the matrix Sp͑2͒. The shear matrices of Eq. ͑1.1͒ can be written as transformation ͓16͔ 1 s Ј ͩ ͪ ϭ ͑Ϫ ͒ v AB v exp isX1 , ͩ 1 ͪ ϭͩ ͪͩ 1 ͪ ͑ ͒ 01 Ј , 4.2 v2 CD v2 ͑A1͒ 10 with ͩ ͪ ϭexp͑ϪiuX ͒, u 1 2 v1 wϭ . ͑4.3͒ v2 with This bilinear representation deals only with the ratio of the 0 i 00 first component to the second in the column vector to which ϭͩ ͪ ϭͩ ͪ ͑ ͒ X1 , X2 , A2 the ABCD matrix is applicable. In polarization optics, for 00 i 0 instance, v1 and v2 correspond to the two orthogonal ele- ments of polarization. which serve as the generators. If we introduce a third matrix
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i 0 In order to transform the above expression into the decom- X ϭͩ ͪ , ͑A3͒ position of Eq. ͑2.1͒, we take the conjugate of each of the 3 0 Ϫi matrices with it generates squeeze transformations: 1 1 i ϭ ͩ ͪ ͑ ͒ C1 . B4 e 0 ͱ2 i 1 ͑Ϫ ͒ϭͩ ͪ ͑ ͒ exp i X3 Ϫ . A4 0 e Ϫ1 Then C1WC1 leads to The matrices X , X , and X form the following closed set 1 2 3 Ϫ Ϫ of commutation relations: cos sin cosh sinh cos sin ͩ ͪͩ ͪͩ ͪ . sin cos sinh cosh sin cos ϭ ϭϪ ͓X1 ,X2͔ iX3 , ͓X1 ,X3͔ 2iX1 , ͑B5͒ ͑A5͒ ϭ ͓X2 ,X3͔ 2iX2 . We can then take another conjugate with
The generators X and X produce the third generator X . 1 11 1 2 3 ϭ ͩ ͪ ͑ ͒ C2 . B6 Then these three generators form a closed set of commuta- ͱ2 Ϫ11 tion relations for the Sp͑2͒ group ͓3,10,22͔. ͑ ͒ Ϫ Ϫ The Sp 2 group can be generated by two seemingly dif- Then the conjugate C C WC 1C 1 becomes ferent sets of generators, namely the shear-squeeze genera- 2 1 1 2 tors of Eqs. ͑A2͒ and ͑A3͒ and the squeeze-rotation genera- cos Ϫsin e 0 cos Ϫsin tors, which are conventionally expressed as ͩ ͪͩ ͪͩ ͪ . ͑B7͒ sin cos 0 eϪ sin cos 1 i 0 1 0 i ϭ ͩ ͪ ϭ ͩ ͪ This expression is the same as the decomposition given in B1 , B2 , 2 0 Ϫi 2 i 0 Eq. ͑2.1͒. ͑ ͒ A6 The combined effect of C2C1 is 1 0 Ϫi ϭ ͩ ͪ i/4 i/4 J , 1 e e 2 i 0 ϭ ͩ ͪ ͑ ͒ C2C1 Ϫ Ϫ . B8 ͱ2 Ϫe i /4 e i /4 when applied to a two-dimensional xy space. The J matrix generates rotations around the origin while B1 and B2 gen- If we take the conjugate of the matrix W of Eq. ͑B1͒ using erate squeezes along the xy axes and along axes rotated by the above matrix, the elements of the ABCD matrix become 45°, respectively. It is clear that one representation can be transformed into the other at the level of generators. The 1 ͑ ͒ ͑ ͒ Aϭ ͑␣ϩ␣*ϩϩ*͒, generators of Eqs. A2 and A3 can be written as 2 ϭ Ϫ ϭ ϩ ϭ ͑ ͒ X1 B2 J, X2 B2 J, X3 2B1 , A7 Ϫi Bϭ ͑␣Ϫ␣*ϩϪ*͒, ͑ ͒ 2 where J, B1, and B2 are given in Eq. A6 . ͑B9͒ i APPENDIX B: BARGMANN DECOMPOSITION Cϭ ͑␣Ϫ␣*Ϫϩ*͒, 2 In his 1947 paper ͓9͔, Bargmann considered 1 ␣ Dϭ ͑␣ϩ␣*ϪϪ*͒. Wϭͩ ͪ , ͑B1͒ 2 * ␣* We can see from this expression that all the elements in the with ␣␣*Ϫ*ϭ1. There are three independent param- ABCD matrix are real numbers. Indeed, the ␣ representa- eters. Bargmann then observed that ␣ and  can be written tion of Eq. ͑B1͒ is equivalent to the ABCD representation, as whose components can be written as
Ϫ ϩ Ϫ Ϫ ␣ϭ͑cosh ͒e i( ), ϭ͑sinh ͒e i( ). ͑B2͒ Aϭ͑cosh ͒cos͑ϩ͒ϩ͑sinh ͒cos͑Ϫ͒,
Then W can be decomposed into BϭϪ͑cosh ͒sin͑ϩ͒Ϫ͑sinh ͒sin͑Ϫ͒, ͑ ͒ Ϫi Ϫi B10 e 0 cosh sinh e 0 Cϭ͑cosh ͒sin͑ϩ͒Ϫ͑sinh ͒sin͑Ϫ͒, Wϭͩ ͪͩ ͪͩ ͪ . 0 ei sinh cosh 0 ei ͑B3͒ Dϭ͑cosh ͒cos͑ϩ͒Ϫ͑sinh ͒cos͑Ϫ͒.
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